BOEING PROBLEM

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BOEING PROBLEM

• Eugene Lavretsky, Boeing
• Heinz Engl
• Alistair Fitt
• Ian Frigaard
• Borislava Gutarts
• Philipp Kuegler
• Xinosheng Li
• Alfonso Limon
• Yajun Mei
• John Ockendon

.The cast in alphabetical order
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PROBLEM

• Assume that we are interested in unmanned
  aircraft only
• Use simplest 6 DoF aerodynamic model
• Restrict motion to 2D
• Try to determine lift, drag, thrust and pitching
  coefficients from (noisy) measurements of
  aircraft position, speed, pitch rate and pitch
  angle

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EQUATIONS
T, L, D, M can be assumed to be the coefficients
that we are after
known (noisy)
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PARADIGM

• For simplicity we will concentrate almost
  entirely on the linear equation

where x is a scalar, a and b are constants and u
is the input, which will be exactly
prescribed. Approximations to a and b will be
denoted by , noisy versions of x(t) will
be denoted by
Note we will not consider the optimal control
problem here.
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On- or Off-line?

• Off-line collect all data from a flight return
  to base.
• This is not wholly desirable as a great deal of
  information must be collected.
• Also does not allow live experiments.
• Adaptive control is online but does not
  generalise to nonlinear equations.Our goal will
  be to minimise predictive rather than tracking
  errors.
• Note that in real life things are really much
  more complicated as the desired coefficients are
  not constant, but depend on the independent
  variables and time.

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TYPICAL OFFLINE METHOD

• Simple regression collect all the data

Then use Moore-Penrose generalised inverse to
write
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FINE TUNING

• VARIOUS ALTERNATIVES
• (i) Calculate the derivatives using eg Euler
• (ii) Integrate both sides and do numerical
  integration
• (iii)Take Laplace transforms
• Note that all except (iii) work just fine for
  nonlinear equations, only the dependence on the
  PARAMETERS must be linear.

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OFFLINE RESULTS

• These can very easily be coded up
• Using Eulers method to approximate derivatives
  is AWFUL.
• Integrating both sides first and doing numerical
  integration is much better,
• Taking Laplace transform is fine too, but these
  are ALL OFFLINE.

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EXAMPLE RESULTS

• Use the problem a -1/2, b 2, u sin(t)
• (nb here the noise is 1 uniform could use
  other sorts)
• (i) Do it all exactly get
• (ii) Do it all exactly but with added noise get
• (iii) Use Eulers method with noise get

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AN ONLINE METHOD

• For
• GENERAL PROCEDURE
• (i) Based on observed
• find an algorithm to estimate a and b
• (ii) For an input u(t) determine PE (Persistence
  of Excitation) conditions so that
• Note that the noise can be added either to x(t)
  or to the differential equation both cases are
  similar but we did not test the latter.

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METHOD
FOR
Suppose for a moment that we know the
xsMINIMISE for a and b to give P G-1B in the
form
Now replace the xs with
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FINE TUNING

• Now note that by doing some integration by parts
  the method becomes

Finally, note that if we know something about the
noise (say it has mean 0 and variance ?2), then
it is better to use
to estimate the square of x(t)
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THE PE QUESTION

• This method is only feasible if the matrix G-1
  exists, so we need
• det(G) ? 0.
• This is assured if and only if we satisfy the
  PE condition (u(t) is rich enough)

This will ensure that as t?? and tend
to the correct values.
NB one way of checking this is to propose
conditions on det(G) as t??
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RESULTS

• Numerical experiments mostly work very well if a
  and b are both O(1).
• (There may be a few starting difficulties to get
  over but these can be sorted out by better
  numerical integration methods.)
• However, there may be wild divergence if a, b
  and/or u are either very large or very small.

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• This procedure is fully online as the integrals
  can be updated by adding only one value
• (NOTE in examples we simply used the trapezium
  rule to do the integrals)

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RESULTS 1 (a possibly difficult example x is a
slave to u)
CASE u 1 a -1, b 1 x 1 e-t Red dots
show successive approximations to the
solutions. The fact that x is a slave to u
suggests that the method might not work but it
does - and Yajun can PROVE it!
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RESULTS 2 (divergence)
CASE u sinh(t) a -1, b 1 x 1 e-t Red
dots show successive approximations to the
solutions. We see divergence, followed by
convergence (to the wrong solution!)
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RESULTS 3 (another possibly difficult example)
CASE u 1/(1t) a -1, b 1 x a mess of
Eis Red dots show successive approximations to
the solutions.
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RESULTS 3 continued
This leads us to consider the relevance of this
method to the ULTIMATE study group problem 0
x(t) bu(t) () The result of attacking
this problem using a gradient method (as in the
book) suggests that u must not decay too fast at
infinity if the PE condition is to be satisfied.
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MORE ABOUT THE PE CONDITION
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PARAMETER IDENTIFICATION

• (i) Traditional approach F(b) x where
• F is the parameter to solution map.
• We want to minimize ??b b?? where b
  contains a priori information. This would lead to
  standard iterative method.
• (ii) All at once approach
• ?? b - btrue ?? should be minimised under the
  constraint
• G(b,x) 0.
• (differential equation as constraint)
• This leads to a saddle point problem, which can
  be solved using mixed finite elements.

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PARAMETER IDENTIFICATION

• (iii) Abstract structure here
• G(b,x,u) 0 treat as constraint.
• ?? b - btrue ?? should be minimised as t ??, n
  ??, or ?
• or we minimise ?? b - btrue ??2 ?? u - uideal
  ??2
• where we have information on u
• (a) u is known (prescribed)
• (b) feasibility constraints on u
• Methods for that could also lead to strategies
  for finding good u, and allow for error analysis.
• (iv) Analogy row-action methods in tomography
  (use information as it comes).

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FINALLY

• 0 x(t) bu(t) ()
• One proposal to solve () is to regularise by
  replacing the LHS by the term ?x(t). Numerical
  experimentation suggests that there is an
  interesting trade-off between the size of ? and
  the behaviour of u.

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THE BOOK

• Applied Nonlinear Control
• J-J E. Slotine, Weiping Li, Prentice Hall 1991
• FINIS

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COLEMANBALLS 1

• SPOOKY FACT OF THE WEEK
• Chris Farmer currently lives in the same house
  that Bill Lionheart grew up in

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COLEMANBALLS 2

• PSYCHIC POWERS DEMONSTRATION OF THE WEEK
• Before you start to tell me how you would do it,
  wait and Ill tell you a better way of doing
  it
• Yajun Mei

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COLEMANBALLS 3

• WORST ESTIMATE OF THE WEEK
• It varies from 15 to 50 so only by a factor of
  5
• John Ockendon

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COLEMANBALLS 4

• NEW LAW OF PHYSICS OF THE WEEK
• Of course its possible for an aircraft to
  accelerate vertically upwards without using any
  thrust
• IAN FRIGAARD

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COLEMANBALLS 5

• THREAT TO US SECURITY OF THE WEEK
• Invasion of the killer worms
• L.A. Times Headline this morning

31
COLEMANBALLS 6

• POLITICAL STATEMENT OF THE WEEK
• The phrase Ill be back is reserved for
  Austrians
• Ottmar Scherzer